Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Threshold circuits of bounded depth
Journal of Computer and System Sciences
On the Size of Weights for Threshold Gates
SIAM Journal on Discrete Mathematics
Perceptrons, PP, and the polynomial hierarchy
Computational Complexity - Special issue on circuit complexity
Circuit Complexity before the Dawn of the New Millennium
Proceedings of the 16th Conference on Foundations of Software Technology and Theoretical Computer Science
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
Problems of Information Transmission
Weights of exact threshold functions
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
SIAM Journal on Computing
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A threshold gate is a linear function of input variables with integer coefficients (weights). It outputs 1 if the value of the function is positive. The sum of absolute values of coefficients is called the total weight of the threshold gate. A perceptron of order d is a circuit of depth 2 having a threshold gate on the top level and conjunctions of fan-in at most d on the remaining level. For every n and d ≤ D ≤ ∈n1/6 we construct a function computable by a perceptron of order d but not computable by any perceptron of order D with total weight 2o(nd/D4d). In particular, if D is a constant, our function is not computable by any perceptron of order D with total weight 2o(nd). Previously functions with this properties were known only for d = 1 (and arbitrary D) [2] and for D = d [12].